Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of

tabletop game Tabletop games or tabletops are games that are normally played on a table or other flat surface, such as board games, card games, dice games, miniature wargames, or tile-based games. Classification according to equipment used Tabletop ga ...

s, including

dice games Dice games are games that use or incorporate one or more dice as their sole or central component, usually as a random device. The following are games which largely, if not entirely, depend on dice: Collectible dice games Patterned after the su ...

board game Board games are tabletop games that typically use . These pieces are moved or placed on a pre-marked board (playing surface) and often include elements of table, card, role-playing, and miniatures games as well. Many board games feature a co ...

s, role-playing games, and

games of chance A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from ...

. A traditional die is a cube with each of its six faces marked with a different number of dots ( pips) from one to six. When thrown or rolled, the die comes to rest showing a random

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

from one to six on its upper surface, with each value being equally likely. Dice may also have polyhedral or irregular shapes, may have faces marked with numerals or symbols instead of pips and may have their

number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...

s carved out from the

material Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geolo ...

of the dice instead of marked on it. Loaded dice are designed to favor some results over others for

cheating Cheating generally describes various actions designed to subvert rules in order to obtain unfair advantages. This includes acts of bribery, cronyism and nepotism in any situation where individuals are given preference using inappropriate crit ...

or entertainment.

History

Dice have been used since before recorded history, and it is uncertain where they originated. It is theorized that dice developed from the practice of fortune-telling with the talus of hoofed animals, colloquially known as

knucklebones Knucklebones, also known as scatter jacks, snobs, astragalus, tali, dibs, fivestones, jacks, or jackstones, among many other names, is a game of dexterity played with a number of small objects that are thrown up, caught, and manipulated in va ...

. The

Egyptian Egyptian describes something of, from, or related to Egypt. Egyptian or Egyptians may refer to: Nations and ethnic groups * Egyptians, a national group in North Africa ** Egyptian culture, a complex and stable culture with thousands of years of ...

game of

senet Senet or senat ( egy, znt, translation=passing; cf. Coptic ⲥⲓⲛⲉ /sinə/ "passing, afternoon") is a board game from ancient Egypt. The earliest representation of senet is dated to E from the Mastaba of Hesy-Re, while similar boards and ...

was played with flat two-sided throwsticks which indicated the number of squares a player could move, and thus functioned as a form of dice. Senet was played before 3000

BCE Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...

and up to the 2nd century CE. Perhaps the oldest known dice were excavated as part of a

backgammon Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back nearly 5,000 years to the regions of Mesopotamia and Pe ...

-like game set at the Burnt City, an archeological site in south-eastern

Iran Iran, officially the Islamic Republic of Iran, and also called Persia, is a country located in Western Asia. It is bordered by Iraq and Turkey to the west, by Azerbaijan and Armenia to the northwest, by the Caspian Sea and Turkmeni ...

, estimated to be from between 2800 and 2500 BCE. Bone dice from

Skara Brae Skara Brae is a stone-built Neolithic settlement, located on the Bay of Skaill on the west coast of Mainland, the largest island in the Orkney archipelago of Scotland. Consisting of ten clustered houses, made of flagstones, in earthen dams ...

, Scotland have been dated to 3100–2400 BCE. Excavations from graves at

Mohenjo-daro Mohenjo-daro (; sd, موئن جو دڙو'', ''meaning 'Mound of the Dead Men';Indus Valley civilization settlement, unearthed

terracotta Terracotta, terra cotta, or terra-cotta (; ; ), in its material sense as an earthenware substrate, is a clay-based unglazed or glazed ceramic where the fired body is porous. In applied art, craft, construction, and architecture, terracotta ...

dice dating to 2500–1900 BCE. Knuck dice Steatite 37x27x21 mm

Games involving dice are mentioned in the ancient Indian ''

Rigveda The ''Rigveda'' or ''Rig Veda'' ( ', from ' "praise" and ' "knowledge") is an ancient Indian collection of Vedic Sanskrit hymns (''sūktas''). It is one of the four sacred canonical Hindu texts ('' śruti'') known as the Vedas. Only one ...

'', '' Atharvaveda,'' ''

Mahabharata The ''Mahābhārata'' ( ; sa, महाभारतम्, ', ) is one of the two major Sanskrit literature, Sanskrit Indian epic poetry, epics of ancient India in Hinduism, the other being the ''Ramayana, Rāmāyaṇa''. It narrates the s ...

'' and Buddhist games list. There are several biblical references to "casting lots" ( ''yappîlū ḡōrāl''), as in

Psalm The Book of Psalms ( or ; he, תְּהִלִּים, , lit. "praises"), also known as the Psalms, or the Psalter, is the first book of the ("Writings"), the third section of the Tanakh, and a book of the Old Testament. The title is derived f ...

22, indicating that dicing (or a related activity) was commonplace when the psalm was composed. Knucklebones was a

game of skill A game of skill or game of wits is a game where the outcome is determined mainly by mental or physical skill, rather than chance. Alternatively, a game of chance is one where its outcome is strongly influenced by some randomizing device, such ...

played in

ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...

; a derivative form had the four sides of bones receive different values like modern dice. Although gambling was illegal, many

Romans Roman or Romans most often refers to: *Rome, the capital city of Italy * Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people, the people of ancient Rome *''Epistle to the Romans'', shortened to ''Romans'', a lette ...

were passionate gamblers who enjoyed dicing, which was known as ''aleam ludere'' ("to play at dice"). There were two sizes of Roman dice. ''Tali'' were large dice inscribed with one, three, four, and six on four sides. ''Tesserae'' were smaller dice with sides numbered from one to six. Twenty-sided dice date back to the 2nd century CE and from Ptolemaic Egypt as early as the 2nd century BCE. historical dice

Dominoes Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also c ...

and

playing card A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a f ...

s originated in China as developments from dice. The transition from dice to playing cards occurred in China around the

Tang dynasty The Tang dynasty (, ; zh, t= ), or Tang Empire, was an imperial dynasty of China that ruled from 618 to 907 AD, with an interregnum between 690 and 705. It was preceded by the Sui dynasty and followed by the Five Dynasties and Ten Kingdom ...

(618–907 CE), and coincides with the technological transition from rolls of manuscripts to block printed books. In Japan, dice were used to play a popular game called sugoroku. There are two types of sugoroku. ''Ban-sugoroku'' is similar to

and dates to the

Heian period The is the last division of classical Japanese history, running from 794 to 1185. It followed the Nara period, beginning when the 50th emperor, Emperor Kanmu, moved the capital of Japan to Heian-kyō (modern Kyoto). means "peace" in Japanese ...

(794–1185 CE), while ''e-sugoroku'' is a

racing game Racing games are a video game genre in which the player participates in a racing competition. They may be based on anything from real-world racing leagues to fantastical settings. They are distributed along a spectrum between more realistic ra ...

Use

Dice are thrown onto a surface either from the hand or from a container designed for this (such as a cup or tray). The face of the die that is uppermost when it comes to rest provides the value of the throw. The result of a die roll is determined by the way it is thrown, according to the laws of

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

. A die roll is made

random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ra ...

by uncertainty in minor factors such as tiny movements in the thrower's hand; they are thus a crude form of

hardware random number generator In computing, a hardware random number generator (HRNG) or true random number generator (TRNG) is a device that generates random numbers from a physical process, rather than by means of an algorithm. Such devices are often based on microscopic ...

. One typical contemporary

dice game Dice games are games that use or incorporate one or more dice as their sole or central component, usually as a random device. The following are games which largely, if not entirely, depend on dice: Collectible dice games Patterned after the su ...

craps Craps is a dice game in which players bet on the outcomes of the roll of a pair of dice. Players can wager money against each other (playing "street craps") or against a bank ("casino craps"). Because it requires little equipment, "street ...

, where two dice are thrown simultaneously and wagers are made on the total value of the two dice. Dice are frequently used to introduce randomness into

s, where they are often used to decide the distance through which a piece will move along the board (as in

and ''

Monopoly A monopoly (from Greek el, μόνος, mónos, single, alone, label=none and el, πωλεῖν, pōleîn, to sell, label=none), as described by Irving Fisher, is a market with the "absence of competition", creating a situation where a speci ...

'').

Construction

Arrangement

Common dice are small cubes, most often across, whose faces are numbered from one to six, usually by patterns of round dots called pips. (While the use of Arabic numerals is occasionally seen, such dice are less common.) Opposite sides of a modern die traditionally add up to seven, requiring the 1, 2, and 3 faces to share a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

. The faces of a die may be placed clockwise or counterclockwise about this vertex. If the 1, 2, and 3 faces run counterclockwise, the die is called "right-handed". If those faces run clockwise, the die is called "left-handed". Western dice are normally right-handed, and Chinese dice are normally left-handed. Die Faces

The pips on standard six-sided dice are arranged in specific patterns as shown. Asian style dice bear similar patterns to Western ones, but the pips are closer to the center of the face; in addition, the pips are differently sized on Asian style dice, and the pips are colored red on the 1 and 4 sides. Red fours may be of Indian origin.

Manufacturing

Non-precision dice are manufactured via the plastic injection molding process, often made of polymethyl methacrylate (PMMA). The pips or numbers on the die are a part of the mold. Different pigments can be added to the dice to make them opaque or transparent, or multiple pigments may be added to make the dice speckled or marbled. The coloring for numbering is achieved by submerging the die entirely in paint, which is allowed to dry. The die is then polished via a

tumble finishing Tumble finishing, also known as tumbling or rumbling, is a technique for smoothing and polishing a rough surface on relatively small parts. In the field of metalworking, a similar process called barreling, or barrel finishing,Degarmo, p. 781. wo ...

process similar to rock polishing. The abrasive agent scrapes off all of the paint except for the indents of the numbering. A finer abrasive is then used to polish the die. This process also produces the smoother, rounded edges on the dice. Precision casino dice may have a polished or sand finish, making them transparent or translucent respectively. Casino dice have their pips drilled, then filled flush with a paint of the same

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

as the material used for the dice, such that the center of gravity of the dice is as close to the geometric center as possible. This mitigates concerns that the pips will cause a small bias. All such dice are stamped with a serial number to prevent potential cheaters from substituting a die. Precision backgammon dice are made the same way; they tend to be slightly smaller and have rounded corners and edges, to allow better movement inside the dice cup and stop forceful rolls from damaging the playing surface.

Etymology and terms

The word die comes from

Old French Old French (, , ; Modern French: ) was the language spoken in most of the northern half of France from approximately the 8th to the 14th centuries. Rather than a unified language, Old French was a linkage of Romance dialects, mutually intellig ...

'' dé''; from

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

datum In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted. ...

'' "something which is given or played". While the terms ''

ace An ace is a playing card, die or domino with a single pip. In the standard French deck, an ace has a single suit symbol (a heart, diamond, spade, or club) located in the middle of the card, sometimes large and decorated, especially in the c ...

'', ''deuce'', ''trey'', ''cater'', ''cinque'' and ''sice'' are generally obsolete, with the names of the numbers preferred, they are still used by some professional gamblers to designate different sides of the dice. ''Ace'' is from the Latin ''as'', meaning "a unit"; the others are 2 to 6 in

. When rolling two dice, certain combinations have slang names. The term ''snake eyes'' is a roll of one pip on each die. The '' Online Etymology Dictionary'' traces use of the term as far back as 1919. The US term ''boxcars'', also known as ''midnight'', is a roll of six pips on each die. The pair of six pips resembles a pair of

boxcar A boxcar is the North American ( AAR) term for a railroad car that is enclosed and generally used to carry freight. The boxcar, while not the simplest freight car design, is considered one of the most versatile since it can carry most ...

s on a freight train. Many rolls have names in the game of craps.

Unicode representation

⚀ ⚁ ⚂ ⚃ ⚄ ⚅ Using Unicode characters, the faces can be shown in text using the range U+2680 to U+2685 or using decimal ⚀ to ⚅.

Loaded dice

A loaded, weighted, cheat, or crooked die is one that has been tampered with so that it will land with a specific side facing upwards more or less often than a fair die would. There are several methods for making loaded dice, including rounded faces, off-square faces, and weights. Casinos and gambling halls frequently use transparent

cellulose acetate In biochemistry, cellulose acetate refers to any acetate ester of cellulose, usually cellulose diacetate. It was first prepared in 1865. A bioplastic, cellulose acetate is used as a film base in photography, as a component in some coatings, and ...

dice as tampering is easier to detect than with opaque dice.

Variants

Polyhedral dice

Various shapes like two-sided or four-sided dice are documented in archaeological findings; for example, from Ancient Egypt and the Middle East. While the cubical six-sided die became the most common type in many parts of the world, other shapes were always known, like 20-sided dice in Ptolemaic and Roman times. The modern tradition of using ''sets'' of polyhedral dice started around the end of the 1960s when non-cubical dice became popular among players of

wargame A wargame is a strategy game in which two or more players command opposing armed forces in a realistic simulation of an armed conflict. Wargaming may be played for recreation, to train military officers in the art of strategic thinking, or to s ...

s, and since have been employed extensively in role-playing games and

trading card games A collectible card game (CCG), also called a trading card game (TCG) among other names, is a type of card game that mixes strategic deck building elements with features of trading cards, introduced with ''Magic: The Gathering'' in 1993. Genera ...

. Dice using both the numerals 6 and 9, which are reciprocally symmetric through rotation, typically distinguish them with a dot or underline.

Common variations

Dice are often sold in sets, matching in color, of six different shapes. Five of the dice are shaped like the

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

s, whose faces are

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

s. Aside from the cube, the other four Platonic solids have 4, 8, 12, and 20 faces, allowing for those number ranges to be generated. The only other common non-cubical die is the 10-sided die, a

pentagonal trapezohedron In geometry, a pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces (i.e., it is a decahedron) which are congruent kites. It can ...

die, whose faces are ten

kites A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...

, each with two different edge lengths, three different angles, and two different kinds of vertices. Such sets frequently include a second 10-sided die either of contrasting color or numbered by tens, allowing the pair of 10-sided dice to be combined to generate numbers between 1 and 100. Using these dice in various ways, games can closely approximate a variety of probability distributions. For instance, 10-sided dice can be rolled in pairs to produce a uniform distribution of random percentages, and summing the values of multiple dice will produce approximations to

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

s. Unlike other common dice, a four-sided (tetrahedral) die does not have a side that faces upward when it is at rest on a surface, so it must be read in a different way. On some four-sided dice, each face features multiple numbers, with the same number printed near each vertex on all sides. In this case, the number around the vertex pointing up is used. Alternatively, the numbers on a tetrahedral die can be placed at the middles of the edges, in which case the numbers around the base are used. Normally, the faces on a die will be placed so opposite faces will add up to one more than the number of faces. (This is not possible with 4-sided dice and dice with an odd-number of faces.) Some dice, such as those with 10 sides, are usually numbered sequentially beginning with 0, in which case the opposite faces will add to one less than the number of faces. Some twenty-sided dice have a different arrangement used for the purpose of keeping track of an integer that counts down, such as health points. These ''spindown dice'' are arranged such that adjacent integers appear on adjacent faces, allowing the user to easily find the next lower number. They are commonly used with collectible card games.

Rarer variations

"Uniform fair dice" are dice where all faces have equal probability of outcome due to the symmetry of the die as it is

face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...

. In addition to the Platonic solids, these theoretically include: *

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan s ...

s, the

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ...

of the 13 Archimedean solids: 12, 24, 30, 48, 60, 120 sides *

Trapezohedra In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...

, the duals of the infinite set of

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

s, with kite faces: any even number not divisible by 4 (so that a face will face up), starting from 6 * Bipyramids, the duals of the infinite set of prisms, with triangle faces: any multiple of 4 (so that a face will face up), starting from 8 *

Disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same sh ...

s, an infinite set of tetrahedra made from congruent non-regular triangles: 4 sides. This is a less symmetric tetrahedron than the Platonic tetrahedron, but still sufficiently symmetrical to be face-transitive. Similarly, pyritohedra and

tetartoid In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

s are less symmetrical but still face-transitive dodecahedra: 12 sides. Two other types of polyhedra are technically not face-transitive, but are still fair dice due to symmetry: *

s: the basis of barrel dice *

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

s: the basis of long dice and teetotums

Long dice Long dice (sometimes oblongFinkel 2004, p 39. or stick dice) are dice, often roughly right prisms or (in the case of barrel dice) antiprisms, designed to land on any of several marked lateral faces, but neither end. Landing on end may be rendered ...

and

teetotum A teetotum (or T-totum) is a form of spinning top most commonly used for gambling games. It has a polygonal body marked with letters or numbers, which indicate the result of each spin. Usage goes back to (at least) ancient Greeks and Romans, w ...

s can in principle be made with any number of faces, including odd numbers. Long dice are based on the infinite set of prisms. All the rectangular faces are mutually face-transitive, so they are equally probable. The two ends of the prism may be rounded or capped with a pyramid, designed so that the die cannot rest on those faces. 4-sided long dice are easier to roll than tetrahedra, and are used in the traditional board games

dayakattai Dayakattai or Dayaboss is a Tamil dice game played by 2 or 4 people (or multiples) by forming teams. It originated in Tamil Nadu (a southern state of India) and is comparable to another dice game from the country called Pachisi. Dayakattai takes ...

and

daldøs Daldøs ø.html"_;"title="al'dø">al'døsis_a_Running-fight_game.html" ;"title="ø">al'døs.html" ;"title="ø.html" ;"title="al'dø">al'døs">ø.html" ;"title="al'dø">al'døsis a Running-fight game">running-fight board game only known from a f ...

Non-numeric dice

The faces of most dice are labelled using sequences of whole numbers, usually starting at one, expressed with either pips or digits. However, there are some applications that require results other than numbers. Examples include letters for

Boggle ''Boggle'' is a word game invented by Allan Turoff and originally distributed by Parker Brothers. The game is played using a plastic grid of lettered dice, in which players look for words in sequences of adjacent letters. Rules One player begi ...

, directions for ''

Warhammer Fantasy Battle ''Warhammer'' (formerly ''Warhammer Fantasy Battle'' or just ''Warhammer Fantasy'') is a tabletop miniature wargame with a medieval fantasy theme. The game was created by Bryan Ansell, Richard Halliwell, and Rick Priestley; it was publishe ...

'',

Fudge dice ''Fudge'' is a generic role-playing game system for use in freeform role-playing games. The name "''FUDGE''" was once an acronym for ''Freeform Universal Donated'' (later, ''Do-it-yourself'') ''Gaming Engine'' and, though the acronym has since b ...

, playing card symbols for poker dice, and instructions for sexual acts using

sex dice Sex dice is a dice game intended to heighten the sexual atmosphere and promote foreplay. Instead of numbers, each face on the dice contains the name of a body part; the body part that faces up when the die is rolled must then be given sexual atte ...

Alternatively-numbered dice

Dice may have numbers that do not form a counting sequence starting at one. One variation on the standard die is known as the "average" die. These are six-sided dice with sides numbered 2, 3, 3, 4, 4, 5, which have the same arithmetic mean as a standard die (3.5 for a single die, 7 for a pair of dice), but have a narrower range of possible values (2 through 5 for one, 4 through 10 for a pair). They are used in some table-top

s, where a narrower range of numbers is required. Other numbered variations include Sicherman dice and

intransitive dice A set of dice is intransitive (or nontransitive) if it contains three dice, ''A'', ''B'', and ''C'', with the property that ''A'' rolls higher than ''B'' more than half the time, and ''B'' rolls higher than ''C'' more than half the time, but it is ...

Spherical dice

Black and red round 6-sided die (cropped)

A die can be constructed in the shape of a sphere, with the addition of an internal cavity in the shape of the

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...

of the desired die shape and an internal weight. The weight will settle in one of the points of the internal cavity, causing it to settle with one of the numbers uppermost. For instance, a sphere with an octahedral cavity and a small internal weight will settle with one of the 6 points of the cavity held downwards by the weight.

Applications

Polyhedral dice are commonly used in role-playing games. The fantasy role-playing game ''

Dungeons & Dragons ''Dungeons & Dragons'' (commonly abbreviated as ''D&D'' or ''DnD'') is a fantasy tabletop role-playing game (RPG) originally designed by Gary Gygax and Dave Arneson. The game was first published in 1974 by Tactical Studies Rules, Inc. (TS ...

'' (D&D) is largely credited with popularizing dice in such games. Some games use only one type, like ''

Exalted ''Exalted'' is a high fantasy tabletop role-playing game originally published by White Wolf Publishing in July 2001. The game is currently in its third edition. It was originally created by Robert Hatch, Justin Achilli and Stephan Wieck, and w ...

'' which uses only ten-sided dice. Others use numerous types for different game purposes, such as D&D, which makes use of all common polyhedral dice. Dice are usually used to determine the outcome of events. Games typically determine results either as a total on one or more dice above or below a fixed number, or a certain number of rolls above a certain number on one or more dice. Due to circumstances or character skill, the initial roll may have a number added to or subtracted from the final result, or have the player roll extra or fewer dice. To keep track of rolls easily,

dice notation Dice notation (also known as dice algebra, common dice notation, RPG dice notation, and several other titles) is a system to represent different combinations of dice in wargames and tabletop role-playing games using simple algebra-like notation su ...

is frequently used. Many board games use dice to randomize how far pieces move or to settle conflicts. Typically, this has meant that rolling higher numbers is better. Some games, such as ''

Axis & Allies ''Axis & Allies'' is a series of World War II strategy game, strategy board games. The first version was initially published in 1981 and a second edition known colloquially as ''Axis & Allies: Classic'' was published in 1984. Played on a board d ...

'', have inverted this system by making the lower values more potent. In the modern age, a few games and game designers have approached dice in a different way by making each side of the die similarly valuable. In '' Castles of Burgundy'', players spend their dice to take actions based on the die's value. In this game, a six is not better than a one, or vice versa. In ''

Quarriors Quarriors! is a pool-building game made by WizKids. In it Players represent a titular mystical warrior, and players roll dice which gives them Quiddity (an in-game currency) which allows them to purchase additional Spell Dice or Creature Dice, whic ...

'' (and its descendant, '' Dicemasters''), different sides of the dice can offer completely different abilities. Several sides often give resources while others grant the player useful actions. Dice can be used for divination and using dice for such a purpose is called cleromancy. A pair of common dice is usual, though other forms of polyhedra can be used. Tibetan Buddhists sometimes use this method of divination. It is highly likely that the

Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...

used the

Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

as dice. They referred to such dice as "the dice of the gods" and they sought to understand the universe through an understanding of geometry in polyhedra. Astrological dice are a specialized set of three 12-sided dice for divination; the first die represents the planets, the Sun, the Moon, and the nodes of the Moon, the second die represents the 12

zodiac signs In Western astrology, astrological signs are the twelve 30-degree sectors that make up Earth's 360-degree orbit around the Sun. The signs enumerate from the first day of spring, known as the First Point of Aries, which is the vernal equinox. ...

, and the third represents the 12 houses. A specialized icosahedron die provides the answers of the

Magic 8 Ball The Magic 8 Ball is a plastic sphere, made to look like an oversized , that is used for fortune-telling or seeking advice. It was invented in 1946 by Albert C. Carter and Abe Bookman and is currently manufactured by Mattel. The user asks a yes– ...

, conventionally used to provide answers to yes-or-no questions. Dice can be used to generate random numbers for use in

password A password, sometimes called a passcode (for example in Apple devices), is secret data, typically a string of characters, usually used to confirm a user's identity. Traditionally, passwords were expected to be memorized, but the large number of ...

s and

cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...

applications. The Electronic Frontier Foundation describes a method by which dice can be used to generate

passphrase A passphrase is a sequence of words or other text used to control access to a computer system, program or data. It is similar to a password in usage, but a passphrase is generally longer for added security. Passphrases are often used to control ...

Diceware Diceware is a method for creating passphrases, passwords, and other cryptographic variables using ordinary dice as a hardware random number generator. For each word in the passphrase, five rolls of a six-sided die are required. The numbers from ...

is a method recommended for generating secure but memorable passphrases, by repeatedly rolling five dice and picking the corresponding word from a pre-generated list.

Notation

In many gaming contexts, especially tabletop role-playing games, shorthand notations representing different dice rolls are used. A "d" or "D" is used to indicate a die with a specific number of sides; for example,d4denotes a four-sided die. If several dice of the same type are to be rolled, this is indicated by a leading number specifying the number of dice. Hence,6d8 means the player should roll six eight-sided dice and add the results. Modifiers to a die roll can also be indicated as desired. For example, 3d6+4 instructs the player to roll three six-sided dice, calculate the total, and add four to it.

References

Bibliography

* Diaconis, Persi; and Keller, Joseph B.
"Fair Dice"
, ''The American Mathematical Monthly'', 96(4):337–339, 1989 (Discussion of dice that are fair "by symmetry" and "by continuity".) * Iverson, G. R.; Longcour, W. H.; et al.; ''Bias and Runs in Dice Throwing and Recording: A Few Million Throws'', Psychometrika, vol. 36, no. 1, March 1971 * Knizia, Reiner, ''Dice Games Properly Explained'', Elliot Right Way Books, 1999,

External links

* analysis of dice probabilities, also featuring Uspenski's work on rolling multiple dice {{Authority control Gaming devices